You're Brothers?? Disparate Corollaries of One Theorem

(1) For each subset A of the circle with measure m, there is a sequence of integers of Beurling-Malliavin density m such that set of the corresponding complex exponentials is complete for L^2(A).
(2) Given an infinite graph, simple random walk on each tree in the wired uniform spanning forest is a.s. recurrent.
(3) We give a theorem that has both these as corollaries.